lcmineqlem2 - Mathbox for metakunt

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Theorem lcmineqlem2 42483

Description: Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.)

**Hypotheses**
Ref	Expression
lcmineqlem2.1	⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥
lcmineqlem2.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
lcmineqlem2.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
lcmineqlem2.4	⊢ (𝜑 → 𝑀 ≤ 𝑁)

**Assertion**
Ref	Expression
lcmineqlem2	⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥))

Distinct variable groups: 𝑘,𝑀,𝑥 𝑘,𝑁,𝑥 𝜑,𝑘,𝑥 Allowed substitution hints: 𝐹(𝑥,𝑘)

**Proof of Theorem lcmineqlem2**
Step	Hyp	Ref	Expression
1		lcmineqlem2.1	. . 3 ⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥
2		lcmineqlem2.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		lcmineqlem2.3	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		lcmineqlem2.4	. . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
5	1, 2, 3, 4	lcmineqlem1 42482	. 2 ⊢ (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥)
6		eqid 2737	. . 3 ⊢ (0[,]1) = (0[,]1)
7		fzfid 13926	. . 3 ⊢ (𝜑 → (0...(𝑁 − 𝑀)) ∈ Fin)
8		0red 11138	. . 3 ⊢ (𝜑 → 0 ∈ ℝ)
9		1red 11136	. . 3 ⊢ (𝜑 → 1 ∈ ℝ)
10		unitsscn 13444	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
11		resmpt 5996	. . . . 5 ⊢ ((0[,]1) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ↾ (0[,]1)) = (𝑥 ∈ (0[,]1) ↦ (𝑥↑(𝑀 − 1))))
12	10, 11	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ↾ (0[,]1)) = (𝑥 ∈ (0[,]1) ↦ (𝑥↑(𝑀 − 1)))
13		nnm1nn0 12469	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
14		expcncf 24903	. . . . 5 ⊢ ((𝑀 − 1) ∈ ℕ₀ → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ∈ (ℂ–cn→ℂ))
15		rescncf 24874	. . . . . 6 ⊢ ((0[,]1) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ)))
16	10, 15	ax-mp 5	. . . . 5 ⊢ ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ))
17	3, 13, 14, 16	4syl 19	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 − 1))) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ))
18	12, 17	eqeltrrid 2842	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝑥↑(𝑀 − 1))) ∈ ((0[,]1)–cn→ℂ))
19		elfznn0 13565	. . . . . 6 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ₀)
20		neg1cn 12135	. . . . . . 7 ⊢ -1 ∈ ℂ
21		expcl 14032	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
22	20, 21	mpan 691	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ → (-1↑𝑘) ∈ ℂ)
23	19, 22	syl 17	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → (-1↑𝑘) ∈ ℂ)
24	23	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (-1↑𝑘) ∈ ℂ)
25	3	nnnn0d 12489	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
26	2	nnnn0d 12489	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
27		nn0sub 12478	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ₀))
28	25, 26, 27	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ₀))
29	4, 28	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
30		nn0z 12539	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℤ)
31	19, 30	syl 17	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℤ)
32		bccl 14275	. . . . . . 7 ⊢ (((𝑁 − 𝑀) ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀)C𝑘) ∈ ℕ₀)
33	31, 32	sylan2 594	. . . . . 6 ⊢ (((𝑁 − 𝑀) ∈ ℕ₀ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℕ₀)
34	29, 33	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℕ₀)
35	34	nn0cnd 12491	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℂ)
36	24, 35	mulcld 11156	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) ∈ ℂ)
37		resmpt 5996	. . . . . 6 ⊢ ((0[,]1) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ↾ (0[,]1)) = (𝑥 ∈ (0[,]1) ↦ (𝑥↑𝑘)))
38	10, 37	ax-mp 5	. . . . 5 ⊢ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ↾ (0[,]1)) = (𝑥 ∈ (0[,]1) ↦ (𝑥↑𝑘))
39		expcncf 24903	. . . . . . 7 ⊢ (𝑘 ∈ ℕ₀ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
40	19, 39	syl 17	. . . . . 6 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ))
41		rescncf 24874	. . . . . . 7 ⊢ ((0[,]1) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ)))
42	10, 41	ax-mp 5	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ))
43	40, 42	syl 17	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ↾ (0[,]1)) ∈ ((0[,]1)–cn→ℂ))
44	38, 43	eqeltrrid 2842	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → (𝑥 ∈ (0[,]1) ↦ (𝑥↑𝑘)) ∈ ((0[,]1)–cn→ℂ))
45	44	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥 ∈ (0[,]1) ↦ (𝑥↑𝑘)) ∈ ((0[,]1)–cn→ℂ))
46	6, 7, 8, 9, 18, 36, 45	3factsumint 42478	. 2 ⊢ (𝜑 → ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥))
47	5, 46	eqtrd 2772	1 ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ↾ cres 5626 (class class class)co 7360 ℂcc 11027 0cc0 11029 1c1 11030 · cmul 11034 ≤ cle 11171 − cmin 11368 -cneg 11369 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 [,]cicc 13292 ...cfz 13452 ↑cexp 14014 Ccbc 14255 Σcsu 15639 –cn→ccncf 24853 ∫citg 25595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-omul 8403 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cn 23202 df-cnp 23203 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-ovol 25441 df-vol 25442 df-mbf 25596 df-itg1 25597 df-itg2 25598 df-ibl 25599 df-itg 25600 df-0p 25647

This theorem is referenced by: lcmineqlem3 42484

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